1

Geometrically Enhanced Thermoelectric Effects in

Graphene Nanoconstrictions

Achim Harzheim,†,Ⅱ Jean Spiece,‡,Ⅱ Charalambos Evangeli,†,‡ Edward McCann,‡ Vladimir

Falko,§ Yuewen Sheng,† Jamie H. Warner,† G. Andrew D. Briggs,† Jan A. Mol,† Pascal Gehring,

†,¶,* and Oleg V. Kolosov,‡, *

† Department of Materials

University of Oxford Parks Road, OX1 3PH Oxford, UK ‡ Department of Physics

Lancaster University Bailrigg, LA1 4YB, Lancaster, UK § School of Physics & Astronomy University of Manchester Oxford Road, M13 9PL, Manchester, UK ¶ Kavli Institute of Nanoscience Delft University of Technology Lorentzweg 1, 2628 CJ, Delft, Netherlands

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KEYWORDS: thermoelectrics, Peltier, Seebeck, graphene nanostructures, scanning thermal microscopy

ABSTRACT. The influence of nanostructuring and quantum confinement on the thermoelectric

properties of materials has been extensively studied. While this has made possible multiple

breakthroughs in the achievable figure of merit, classical confinement and its effect on the local

Seebeck coefficient has mostly been neglected, as has the Peltier effect in general due to the

complexity of measuring small temperature gradients locally.

Here we report that reducing the width of a graphene channel to 100 nanometers changes the

Seebeck coefficient by orders of magnitude. Using a scanning thermal microscope allows us to

probe the local temperature of electrically contacted graphene two-terminal devices or to locally

heat the sample. We show that constrictions in mono- and bilayer graphene facilitate a spatially

correlated gradient in the Seebeck and Peltier coefficient, as evidenced by the pronounced

thermovoltage 𝑉 and heating/cooling response Δ𝑇 respectively. This geometry dependent

effect, which has not been reported previously in 2D materials, has important implications for

measurements of patterned nanostructures in graphene and points to novel solutions for effective

thermal management in electronic graphene devices or concepts for single material thermocouples.

TEXT. Solid-state thermoelectric devices have long been attractive to researchers and engineers

alike, due to their capability of reliably converting waste heat to electricity and the possible thermal

management applications.1–4 In addition, an in-depth understanding of thermoelectric phenomena

is important to correctly interpret photocurrent and electrical transport measurements where these

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phenomena can play a major role.4,5 There are two complementary thermoelectric effects, the

Seebeck effect and its Onsager reciprocal, the Peltier effect. For the first, a temperature difference

Δ𝑇 will induce the buildup of a thermovoltage Δ𝑉 𝑆∆𝑇 across a material with a Seebeck

coefficient S. Vice versa, for the second, an electrical current 𝐼 induces a heat flow 𝑄 Π𝐼, where

Π 𝑇𝑆 is the Peltier coefficient.6

A resurge in interest in this topic was initiated by Hicks and Dresselhaus’ theoretical findings

that reducing the dimensionality of thermoelectric materials could significantly increase their

efficiency.7,8 This is measured by the dimensionless figure of merit 𝑍𝑇 𝑇 – a function of the

electrical 𝜎 and thermal 𝜅 conductivity – and the principle has since been demonstrated by

various groups.9,10 Amongst the techniques that have been employed are building nanocomposites

from nanocrystal blocks,11 nanostructuring quantum dot superlattices,9 the exploitation of negative

correlations between electrical and thermal conductivity,12 and band engineering.13,14 Moreover,

classical rather than quantum confinement has been reported to cause an increase in the Seebeck

coefficient in gold and Antimony Telluride nanowires.15,16

Here we present high resolution Scanning Thermal Microscopy measurements of 100 nm wide

graphene bow-tie nanoconstrictions that show a pronounced spatial dependence of the Seebeck

and the Peltier effect. This change in the local Seebeck coefficient is attributed to a shortened

effective Electron Mean Free Path (EMFP) due to edge scattering and opens up the possibility to

readily produce two dimensional one-material thermocouples as well as accessible local

temperature management and improved heat dissipation.

We perform our measurements with a Scanning Thermal Microscope (SThM) – effectively an

atomic force microscop (AFM) with a microfabricated resistor incorporated in the tip17 - using two

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different protocols to map the Seebeck and Peltier effect as well as the Joule heating. In the Peltier

measurement, we use a recently developed non-equilibrium scanning probe thermometry

method:18 an AC bias Vbias applied to the device through the global contacts causes an AC current

Ibias which results in Joule heating and Peltier heating/cooling. By measuring the temperature

response of our SThM tip as it is scanned over the AC biased sample and modulating it at the first

(Peltier) and second (Joule) harmonic it is possible to decouple the two effects and extract the

respective heating/cooling values (see Figure 1a for the measurement schematics). In contrast, for

the thermovoltage or Seebeck measurement, the SThM tip is heated by applying a high AC voltage

to it and the global voltage drop over the device is recorded at the second harmonic as the hot tip

is scanned over the sample. Both single layer and multilayer graphene are measured, but no

thickness dependence in the size and distribution of the signal is found.

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Figure 1. Nanoscale mapping of the Peltier effect in graphene nanoconstrictions. (a) An AC voltage bias 𝑉 at 𝑓 induces an AC current 𝐼 through the constriction (black lines). In addition, a low AC bias with a DC offset is applied to the SThM tip through a Wheatstone bridge (magenta line). During scanning, the resulting signal in the tip (red lines) is demodulated at the respective frequency. This thermal signal is then demodulated at the first (𝑓 and second (2𝑓 harmonic, providing the Peltier heating/cooling (green) in b and Joule heating (red) in c, respectively. The blue dashed lines symbolize the reference signal lines. (b) Peltier effect map showing the main heating/cooling effects around the constriction (c) Joule heating map, showing the hot spot in the middle of the constriction. (d) Simultaneously recorded height map used to outline the position of the constriction in the Peltier heating/cooling and Joule heating images. In b and c, the dotted-dashed lines indicate the contact position and the dashed line the outline of the graphene constriction. All scale bars are 1 μm.

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Peltier and Joule heating maps of the bow tie device, are shown in Figure 1b and c, respectively,

where both show a high spatial dependence, with a strong signal around the constriction. The Joule

heating exhibits a temperature increase while the Peltier signal shows cooling/heating on the

respective side of the constriction and a node in the middle. The Peltier signal shown here

importantly corresponds to the measured amplitude multiplied by the sine of the phase signal. It is

the temperature at a certain phase at the maximum applied modulation voltage, since in time

average no discernible Peltier heating or cooling is taking place at the constriction for an AC bias.18

Figure 1d shows the simultaneously measured height signal, which was used to determine the

exact position of the device indicated in b and c.

The Joule heating showing a maximum in the constriction is expected due to the increased local current

density,19 however, given the continuous composition of the material in the constriction area, all

thermoelectric effects in the device would be expected only in the vicinity of the Au electrodes.20 As can be

seen in Figure 1b, the Peltier signal Δ𝑇 becomes strongest around the constriction itself and outlines

the shape of the graphene bow-tie where the signal at the edges is broadened out due to heat spreading into

the surrounding SiO substrate. The SThM measurement of the device without current excitation, also

observed by Tovee et al,21 show that the heat dissipation from the heated tip in the areas with and without

graphene differs by less than 5% (see SI section 7). These suggest that it is reasonable to assume that SiO2/Si

substrate plays dominant role in the heat spreading. The Peltier effect results in heating and cooling of up

to Δ𝑇 ±2K on either side of the constriction for an applied current of 𝐼 90μA. A markedly

similar behavior was found for 𝑉 in the thermovoltage measurements on the same device (see Figure S5

in the Supporting Information) under open-circuit condition, confirming that the signal likely stems from a

changed local Seebeck coefficient. In addition, we observe comparatively weak “conventional”

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Peltier heating/cooling in the vicinity of the Au contacts (see Figure 1b) which is explained by the

formation of a Peltier junction between gold and graphene at the contacts as reported previously.4,20

Such a geometrical modification of the local Seebeck coefficient has been seen in metallic thin-film

stripes and Au nanowires and was explained by structural defects and the metal grain structure, which in

turn reduce the EMFP.15,22 The EMFP of graphene at room temperature, is typically on the order of 100s

of nanometers and thus higher than in gold.23 However, it gets substantially reduced by defect potentials

such as ones stemming from rough edges,24 which in our case have been created by the device patterning

and amount up to an 80% reduction.25 This edge scattering becomes more dominant as the width of the

graphene stripe Δ𝑦 𝑥 reduces, giving a position dependent mean free path, which can be written as

𝑙 𝑥 𝑙 1 𝑐 , (1)

where 𝑙 is the bulk mean free path and 𝑐 and 𝑛 are numerical coefficients specifying the transport

mode and the influence of scattering on the mean free path (see section 11 in the Supporting Information).

To extract the bulk mean free path we perform gate conductance measurements on 43 μm long and 3 μm

wide graphene ribbons that give us 𝑙 226 20 nm (see section 3 Supporting Information).

Using the Mott formula 𝑆 | we obtain an expression for the thermopower as a

function of constriction width (see section 11 in the Supporting Information for more information):

𝑆 1 𝑛 𝑈 𝑛 1 𝑈 , (2)

where 𝑈 | is the exponent of any power law dependence of the EMFP on energy. We

expect this value to be between the short range disorder or electron-phonon interaction value 𝑈 1

and the long range Coulomb interaction 𝑈 1. 26,27 Equation (2) predicts that the local Seebeck

coefficient decreases when the width of the channel is reduced, which leads to regions with different

effective Seebeck coefficients in the bow-tie shaped devices.

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Figure 2. Modelling and fitting of Joule heating and thermoelectric effects in a bow-tie device. (a) From the top: schematic of the tip movement, 1D section cuts through the middle of the constriction of the calculated Seebeck coefficient, the tip-defined moving thermal gradient and the resulting thermovoltage measured and calculated respectively. The inset shows the quadratic tip voltage dependence of the thermovoltage signal in a log-log plot. (b) Joule heating at different applied voltage biases experimentally recorded (dots) and fitted to a COMSOL model (lines). The smallest Joule heating signal (1 Vpp, yellow) is used to extract the electrical and thermal conductivities for the entire model ( 𝜅 120 Wm K , 𝜎 5 ∙ 10 Sm ). (c) Peltier heating/cooling at 1 Vpp, experimental and simulated from the COMSOL model using the calculated Seebeck coefficient from (a). The zero of the tip position is centered at the middle of the constriction for all figures.

Using Equation (1) and (2), we can model 𝑉 𝑥 and compare it to the measured thermovoltage 1-D

line section signals. As shown in Figure 2a, 𝑉 𝑥 𝑆∇𝑇 𝑑𝑥 is calculated by taking the integral

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of 𝑆∇𝑇 over the whole length of the device at each point. In the measurement and in our calculations, the

Seebeck coefficient is only dependent on the width of the constriction and its distribution does not change

as we move the tip, while the temperature gradient Δ𝑇 induced by the heater voltage 𝑉 is always

centered at the tip position x and thereby moves as we scan over the sample. The heater temperature Δ𝑇

is obtained from calibrating the tip and measuring the thermal resistance between the heater and the sample

(see Supporting Information 7). It is worth noting here that there is an inherent uncertainty of the heater

temperature that can lead to an over or underestimation of the measured effect, however this does not

change the conclusion and main results of our work. Fitting the calculated values to the line cut of the

thermovoltage measured with and estimated Δ𝑇 18 2 K gives the dimensionless parameters 𝑐

0.56, 𝑛 2.6 and 𝑈 0.88. Using these fitting values we calculate a bulk Seebeck coefficient of 𝑆

118 μV K , which is similar to previous measurements in graphene at room temperature.28 The Seebeck

coefficient in our model reduces from the bulk value of 𝑆 118 𝜇𝑉𝐾 to 𝑆 0.34 𝜇𝑉𝐾 due to

the reduction of the mean free path within the constriction. This orders of magnitude decrease is due to

Equation (2) involving a difference of terms, resulting in a dramatically varying S for relatively small

changes in the EMFP.

To further test the influence of geometrical confinement on the thermoelectric properties of

graphene devices we have tested an “island” structure, where wide and narrow parts of graphene

alternate and which is showing a pronounced signal at these junctions (see Figure S13 in the

Supporting Information). It is worth to mention that applying a back gate voltage enables us to

change the doping from 𝑝 -doping (-30V) to 𝑝-doping (30V) which results in a modification of

the signal strength in the constriction by approximately 20% due to the changed carrier density

(see Figure S11 in the Supporting Information).28

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The spatially dependent Seebeck coefficient extracted from the thermovoltage fit can be used to develop

a COMSOL model that can reproduce our experimental Joule heating and Peltier signal (see Figure 2b and

c). In this model the effective thermal conductivity 𝜅 120 W mK and the electrical conductivity

𝜎 5 ∙ 10 Sm are the only fit parameters with the spatial heat distribution determined mainly by the

SiO layer and only slightly modified by the fitted value of the thermal conductivity of a single graphene

layer.

We have in addition studied the current dependence of all measured signals by placing the tip on

one side of the constriction as the current through the device, 𝐼 , (in the Peltier and Joule heating

measurements) or through the tip, 𝐼 , (in the thermovoltage measurements) is increased. In

both the Joule heating and the thermovoltage measurements, a square current dependence on the

current is observed (see Figure 2a and b inset), in agreement with the Joule-Lenz law (𝑃 ∝ 𝐼 𝑅).

However, in the Peltier measurement of the bow-tie device, we find that an initially linear

dependence changes to a cubic one as we increase the current 𝐼 . As can be seen in Figure 3a,

the data can be fitted with a combination of a cubic and linear term, where the crossover point is

located at approximately 𝐼 33 μA. This is a deviation from the simple linear dependence

predicted by 𝑄 Π ∗ 𝐼 𝑆𝑇 ∗ 𝐼 . We find this behavior in all geometries measured, with

the crossover happening at different current levels (Figure S7 in the Supporting Information).

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Figure 3. Deviation of the experimental data from the linear Peltier model. (a) Fit of the current dependency of the Peltier heating in the constriction for a linear and cubic (grey line) and fifth order term (red line). For the cubic dependency, which seems to fit the data better, the Peltier heating switches over from a linear to a cubic current dependency where the switchover point is marked by the black dotted-dashed line. The orange dotted line is linear with respect to the current and the blue dotted line is cubic and serve as a guide to the eye. (b) Comparison of the Peltier heating/cooling to the COMSOL model at 3𝑉 . A big discrepancy between the COMSOL model and the experimental data is visible both in shape and in amplitude. The asymmetry in the experimental data might be linked with the nanoscale asymmetry of the nanoconstriction. The inset shows the current dependency of the simulated Peltier heating, which is linear, save for a small correction (∝ 1.02 due to the Joule heating.

We attribute the unusual current dependence observed in our experiments to an “electron wind”

effect: if the drift velocity 𝑣 becomes comparable to the Fermi velocity 𝑣 heat is shifted with

respect to the position of the constriction, effectively cooling one side and heating the other side.

For this effect, we expect the Peltier heating/cooling to take the form of a sum of the common

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linear Peltier effect and a cubic term. Here, in addition to the expected linear Peltier term, there is

a contribution from an increasing drift velocity (linearly increasing with current) and the

temperature of the hot carriers created by Joule heating (quadratic current dependency) that adds

up to an additional cubic term. Indeed, we find that a fit of this model to the measured data provides

a good agreement, compared to other higher order terms (see Figure 3 and SI). The drift velocity

in our devices which is given by 𝑣 0.25 ⋅ 10 ms with 𝐼 40 μA the current

through the device, 𝑛 the carrier density, 𝑒 the elementary charge and 𝑊 100 nm the width of

the constriction is approaching the Fermi velocity in graphene, 𝑣 10 ms . As we do not

observe any significant electrical nonlinearity in the I-V traces, the carrier density is well

approximated by the low current value of 𝑛 10 m . A similar electron wind effect has been

observed for varying gate voltages in graphene devices.29

Another source for this discrepancy, the temperature dependence of the Seebeck coefficient and

its increase caused by Joule heating would give a fifth order dependence since 𝑄 ∝ 𝑆𝑇𝐼 ∝ 𝑇 𝐼 ∝

𝐼 . We have investigated this explanation but found that it does not agree well with the measured

data. Since the measurements are performed at room temperature (300K) and only a few Kelvin

temperature increase due to Joule heating are measured the impact on the Peltier heating/cooling

is negligible (see section 1 in the Supporting Information). The temperature increase is also taken

into account in the COMSOL calculation of the Peltier heating/cooling, which solves for the full

thermoelectric equation ( 𝜌𝐶 𝒖𝛻𝑇 𝛻 𝑘𝛻𝑇 𝑃𝑱 𝑄 , see also Supporting Information but

COMSOL modeling results in only a small deviation of about 2% from the linear exponent (see

inset Figure 3b), which is about two orders lower than observed in the experiment.

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While a heatdrift in the constriction due to high drift currents agrees well with our data and

explains the observed deviation from a linear current dependency expected from the Peltier effect,

other higher order causes are possible and further investigation of this effect will be necessary.

To summarize, we observe a strong geometrical dependence of both the Peltier and the Seebeck

effect in graphene nanoconstrictions dominating over the previously reported thermoelectric effect

at the graphene-metal interface.4,20 We can explain this local variation of the Seebeck coefficient

by a reduction in the EMFP, which is caused by the increased scattering from the edges. Compared

to Au nanowires that have shown a similar effect previously,15 graphene offers a more

straightforward method of influencing the mean free path, due to its lower dimensionality and also

comparatively bigger electron mean free path. Furthermore, we observe an additional contribution

to the Peltier effect by an ‘electron wind’ resulting from the high drift velocity of charge carriers

in the constriction. This work highlights the major influence of disorder and geometry on

thermoelectric properties of graphene. Thus, thermoelectric effects are likely present in graphene

whenever edge scattering becomes appreciable and can lead to undesired heating/cooling.

Similarly, any temperature gradient across an edge scattering region will create a parasitic voltage

drop over the device. These are important consideration for future photothermoelectric as well as

thermal and electrical transport measurements in nanoscale electronic devices.

In addition, our findings have implications for thermal management in future integrated circuits

made out of graphene: The results open a path to producing a single material thermocouple or

Peltier element that can be precisely positioned using electron beam lithography. As shown in

Figure S13 in the Supporting Information, a substantial reduction of the channel width effectively

creates a highly localized Peltier element which could be used for local cooling or temperature

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sensing. Such all-graphene thermocouples could be integrated into planar device structures on a

wafer scale and at comparatively low costs.

Methods

Device fabrication

The devices were fabricated by transferring two different types of CVD graphene,30 multilayer (2-4

layers) and single layer (see Supporting Information), on top of a Si chip with a 300 nm SiO2 and pre-

patterned Cr/Au contacts using a standard wet transfer method.31 Subsequently, the graphene was

patterned into the different geometries employing standard electron-beam lithography and then

etched into different geometries using oxygen plasma etching.

Scanning Thermal Microscopy measurement methods

The SThM is located in a high vacuum environment, prohibiting parasitic heat transfer between the

tip and the sample to achieve a better thermal resolution.18,32 In our measurements, the spatial

resolution is limited by the size of the tip-sample contact which is on the order of tens of

nanometers.

We used two distinct scanning measurement methods, passive SThM temperature probing and

active heated-probe local thermovoltage measurements.

In the Peltier measurement, the device is electrically excited with an AC bias 𝑉 through the

global contacts at a frequency of 𝑓 = 17Hz. The SThM tip is scanned over the sample, measuring

the temperature Δ𝑇 at the first harmonic (𝑓 ) using a SRS830 lock-in (see Figure 1a).

Simultaneously the unmodulated temperature-dependent DC signal and the Joule heating signal

Δ𝑇 , measured at the second harmonic (2𝑓), are recorded. The Peltier and Joule measurements

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were performed following Menges et al.,18 to exclude tip-sample contact-related artefacts (see

section 7 in the Supporting Information and 18).

In contrast, for the thermovoltage scanning method, the SThM tip is heated up by applying a

high AC voltage of 𝑉 2.24 𝑉 to the temperature sensor. This Joule heating of the SThM

tip at a frequency of 𝑓 = 57Hz, results in a modulation of the SThM resistor temperature of

approximately 60K, leading to a SThM tip temperature modulation of ΔT 18 2 K at the

interface with graphene (see section 7 Supporting Information). This local heat source is then

scanned over the sample while the global voltage drop 𝑉 over the two contacts is measured with

a SR560 voltage pre-amplifier and a SRS830 lock-in amplifier at the second harmonic (2𝑓 ) (see

Figure 1b). Our thermovoltage measurements do not require electrical contact between the tip and the

sample, as does a similar method reported previously,33 and thereby eliminate linked uncertainty, as well

as requirements on the strength of the electrical tip-sample contact. To rule out effects on the measured

signal stemming from accidental phase errors in the lock-in signal, we performed a DC

measurement where a positive and negative square wave are applied respectively and the two

resulting temperature maps are subtracted. This configuration shows the same signal as the AC

measurements, thereby eliminating the possibility of an unintended phase effect causing the signal

(see Supporting Information).

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ASSOCIATED CONTENT

Supporting Information.

The Supporting Information is available free of charge from the ACS website and DOI.

Peltier correction terms, graphene characterization, EMFP evaluation, I-V traces, DC measurement,

thermovoltage measurements, SThM setup, FE analysis, current dependencies other geometries, long

ribbon geometry, spatially dependent Seebeck coefficient theory, gate dependency, heat movement by

carriers, island geometry measurements, sample preparation. (PDF)

AUTHOR INFORMATION

Corresponding Author

*Email: p.gehring@udelft.nl; o.kolosov@lancaster.ac.uk

Author Contributions

Ⅱ A. H. and J. S. contributed equally to this work.

A.H. fabricated the devices. A.H., P.G. and O.K. drafted the manuscript. A.H. J.S., C.E. and P.G.

performed the measurements. A.H. J.S., C.E., P.G., J.M. and O.K. analyzed and processed the

data. Y.S. and J.W. provided the graphene. E.M. carried out the theoretical modelling and J.S.

performed the FEM calculations. O.K. and P.G. supervised, conceived and designed the

experiments. All authors discussed the results and contributed to manuscript revision. ‡These

authors contributed equally. (match statement to author names with a symbol)

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Funding Sources

This work is supported by the UK EPSRC (grant nos. EP/K001507/1, EP/J014753/1,

EP/H035818/1, EP/K030108/1, EP/J015067/1, and EP/N017188/1).

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENT

The authors would like to thank Prof. Colin Lambert and Dr. Hatef Sadeghi for helpful discussions

and Jasper Fried for performing the long ribbon conductance measurements. P.G. acknowledges a

Marie Skłodowska-Curie Individual Fellowship under grant TherSpinMol (ID: 748642) from the

European Union’s Horizon 2020 research and innovation programme. O.K. acknowledges the

EPSRC project EP/K023373/1 and EU project No.604668 QUANTIHEAT.

ABBREVIATIONS

SThM, scanning thermal microscope; EMFP, electron mean free path; AFM, atomic force

microscopy.

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(32) K. Kim, W. Jeong, W. Lee, P. Reddy, ACS Nano 2012, 6, 4248.

(33) B. Lee, K. Kim, S. Lee, J. H. Kim, D. S. Lim, O. Kwon, J. S. Lee, Nano Lett. 2012, 12,

4472.

1

Supporting Information: Geometrically Enhanced

Thermoelectric Effects in Graphene

Nanoconstrictions

Achim Harzheim, Jean Spiece, Charalambos Evangeli, Edward McCann, Vladimir Falko,

Yuewen Sheng, Jamie H. Warner, G. Andrew D. Briggs, Jan A. Mol, Pascal Gehring,* and Oleg

V. Kolosov*

 

 

 

 

This supporting information contains:  

Supplementary figures S1‐S12 and supplementary texts chapter 1‐13 

 

 

 

2

1. Peltier correction due to Joule heating 

Since the Peltier coefficient is proportional to the temperature squared, the Joule heating in our devices undeniably has an 

influence on the measured Peltier heating/cooling. From the measurements of Joule heating in our bow tie devices, an 

increase  of Δ𝑇 10 K can  be  extracted.1  Using  this  value, we  can  estimate  the  expected  contribution  of  Joule 

heating to the Peltier heating/cooling measured. The heat generated or taken away by the Peltier effect is given as 

𝑄 Π𝐼 𝑆𝑇𝐼 𝛽𝑇 𝐼,          (S1) 

Where 𝛽 1 𝑛 𝑈 𝑛 1 𝑈  is the part of the Mott formula independent of the temperature. Then 

assuming 𝑇 𝑇 Δ𝑇  with 𝑇 300𝐾 being the room temperature and Δ𝑇 10 K the maximum increase 

in  temperature  due  to  Joule  heating,

𝑄 𝛽 𝑇 Δ𝑇 𝐼 𝛽 𝑇 𝑇 Δ𝑇 Δ𝑇 𝐼. 

We can then calculate the percentage increase of heat and thereby the temperature due to the correction for the Joule 

heating as follows: 

Δ𝑇 6.8% 

                          (S2) 

 

2. Graphene characterization 

In order to characterize the graphene, Raman spectroscopy and SEM imaging was employed. Pre‐patterning SEM images 

showed continuous coverage and a domain size of roughly 10 μm while the Raman spectroscopy allowed us to conclude 

the number of  layers and quality of  the graphene employed. As shown  in Figure S1a,  the ratio of  the G to 2D peak 

indicates single layer graphene, with a certain degree of disorder not unusual for CVD graphene as is evident by the visible 

D peak. The graphene shown in Figure S1b on the other hand is clearly multiple layers as the G peak is higher than the 2D 

peak. 

3

 

Figure S1. Raman spectroscopy of the graphene used. (a) 1-2 layer graphene and (b) few layer graphene 

 

3. Evaluation of the electron mean free path  

In order to extract the bulk EMFP 𝑙  for the left and right reservoir, long 3 μm wide ribbons are used, which are the same 

width as the reservoirs and therefore give a good estimate. The bulk EMFP 𝑙  can be calculated from a gate trace, where 

the current through the ribbon at a constant voltage  is measured as the gate  is swept.2  In graphene, a curve with a 

minimum at the Dirac point is expected, as shown in Figure S2a.  

Using the transconductance from this measurement, 𝑑𝐺/𝑑𝑉 , to estimate the carrier mobility gives: 

𝜇  ,                                           (S3) 

where 𝐿 43 μm is the ribbon length, 𝐶  is the back gate capacitance (𝜖 3.9 and 𝑑 300 nm for our samples) and 𝑊 3 μm the width of the ribbon.3  We can then calculate the mean free path with Equation S3 from  

𝑙ℏ

𝜇√𝑛𝜋 ,                                         (S4) 

with the carrier concentration 𝑛  given by the position of the dirac point 𝑉 . Six different ribbons (one 

ribbon is displayed in Figure S2 a, b and c) were measured that gave an average of 𝜇 8700 100 𝑐𝑚 𝑉𝑠  and 

𝑙 226 20 nm at 𝑉 0.  

4

 

  

Figure S2: Bulk EMFP extracted from long ribbon back-gate dependent gatetraces. (a) Representative gate dependent current

measurement of a device as a function of the carrier density. (b) calculated EMFP as a function of the carrier density using the

data in a and Equation S4. (c) Extracted mobility 𝜇 as a function of carrier density using the data in a and Equation S3.

 

 

4. I‐V characteristics 

All of the devices measured show linear or minimal deviation from linear I‐V characteristics within the probed 

voltage bias as is shown in Figure S3. This rules out another possible reason for the observed Peltier effect, a 

highly  resistant  bridge or mobility  gap due  to  the  constriction  that would  induce  a  change  in  the  Seebeck 

coefficient  as  predicted  previously.4  Such  a  mobility  band  gap  would  be  reflected  in  a  nonlinear  I‐V 

5

dependence at low currents that is not observed in our measurements. In addition, as we are measuring in 

the  linear  I‐V  regime V  and  I  differ  by  a  constant multiplier,  the  resistance  and  nonlinear  effects  do  not 

influence  the measurements.  The  voltage  applied during Peltier measurements  is  typically 𝑉 2𝑉  for 

the  shown  samples,  corresponding  to  a  current  of 𝐼 25 30μA for  multi‐structured  devices  and 

around 𝐼 30 90μA   for  the  bow  tie  devices. We  have  similarly  recorded  IV  curves  of  our  bow‐tie 

devices at high applied DC voltages. These devices are showing slight non‐linear behavior at higher voltages, 

but  are  still  well  approximated  as  being  in  the  linear  regime  and  in  addition,  show  a  symmetric  current 

response. A typical device IV‐curve is shown below in Figure S3 d. All device resistances were on the order of 

kΩs and were biased at sufficiently low voltages to exhibit linear IV characteristics. 

 

Figure S3: AC I-V traces for the different devices shown in this paper in a,b,c and a typical DC trace seen in our devices in d. (a)

6

Device 3 shown in Figure 2 and 3. (b) Device 1 shown in Figure 1. (c) long wire device show in Figure S8. (d) DC IV trace going

to high currents and biases. While the current does not quite stay linear, no pronounced asymmetry is visible.

5. DC thermal measurements  

 In order to eliminate any  interpretation of measured phase as the origin of the observed Peltier heating and cooling 

effects, a quasi‐DC measurement of the Joule heating is performed. A step function at 1 V  is applied to the device at a 

low  frequency  while  the  temperature  response  is  recorded.  This  is  a  pure  temperature  recording without  a  phase 

dependency to separate Peltier heating/cooling and Joule heating. Subsequently, a second measurement with  1 V  is 

performed  and  subtracted  from  the  first  measurement.  Since  the  Joule  heating  is  symmetric  with  respect  to  the 

constriction, subtracting the two measurements eliminates the Joule heating signal. The Peltier heating on the other hand 

is asymmetric and the result is a purely Peltier heating/cooling map that then has to be divided by two. As can be seen in 

Figure S4, the resulting DC Peltier signal reproduces the AC measurements well. 

Figure S4: DC measurement of a bow tie junction. The image is created by subtracting a positive and a negative image. The

scale bar is 500 nm.

 

 

6. Bow tie structure ‐ thermovoltage measurements 

The thermovoltage of the bow tie device in Figure 1 was measured with the hot (18 K heater excess temperature) tip 

scanning over the device as explained in the experimental section of the main manuscript and in section 7 below.5 For the 

thermovoltage measurements we observe the same features as in the Peltier measurements, with a positive and negative 

thermovoltage on the respective sides of the junction and a zero signal node in between (see Figure S5). This is a further 

7

confirmation that it is a change in the Seebeck coefficient rather than a current induced effect that is responsible for the 

observed signal. 

 

Figure S5: Thermovoltage measurement of Device 1. The thermovoltage map resembles largely the Peltier map. The scale bar is

500 nm.

 

7. SThM measurement setup 

In order to gain insights into a sample’s heat dissipation properties, we deploy Scanning Thermal Microscopy 

(SThM) that uses a standard AFM feedback loop to maintain a constant force between the sample and the 

probe.6 Acting as both a sensor and a heater, the SThM probe consists of a two‐legged Si cantilever with a 

doped Si heater above a conical tip.  

 

Prior to measurement, the probe electrical resistance is calibrated as a function of the temperature and the 

probe excess temperature arising from self‐heating of the heater is quantified for a given electrical power.7 

This also allows us to quantify the thermal resistance of the cantilever Rp. We obtained values of Rp between 

6x104 to 1.2x105 KW‐1 for the probes used in these experiments.  

 

During  thermometry  experiments,  the  probe  resistance  is  monitored  via  a  modified  Wheatstone  bridge 

whose output  is amplified before being  fed  into a  lock‐in amplifier  (Stanford Research Systems, SR‐830). A 

combined AC+DC bias voltage is applied on the bridge. The AC component (91 kHz) provides high sensitivity 

through  lock‐in detection and  the DC part  creates  Joule heating  in  the probe heater. We used a DC offset 

8

generating 26 K excess temperature with respect to the surrounding. When the probe is brought into contact 

with the sample surface, it cools down (heats up) depending on the increase (decrease) of the heat flux to the 

sample  and  consequently  on  the  sample  local  heat  transport  characteristics.  The  variations  of  the  probe 

heater  temperature  generate  variations  of  the  probe  electrical  resistance,  which  are  detected  by  the 

electronics.  Obtaining  temperature  maps  of  operating  devices  can  be  challenging  as  the  probe‐sample 

contact varies and creates artefact in the temperature measurement. To counter such effects, Menges et al.1 

developed a method  taking  into account  the heat  flux  from a heated tip  to  the sample and correcting  the 

measured  temperature  map.  This  is  realized  by  creating  an  oscillating  temperature  field  in  the  sample 

through  an  applied  AC  bias  on  the  device.  The  AC  temperature  variations  can  be  detected  by  the  SThM 

sensor and amplified by lock‐in detection. Then a corrected temperature map is obtained by:1  

𝛥𝑇 𝛥𝑇                                  (S5) 

where 𝛥𝑇  is  the  sample  temperature  rise, 𝛥𝑇  is  the  SThM  heater  excess  temperature  before 

contacting the sample, 𝛥𝑉  is the variation of the SThM signal measured at the device AC driving frequency 

and 𝛥𝑉  is  the  variation  of  the  SThM  signal  sensitive  to  tip‐sample  heat  fluxes.  To  measure  Peltier 

temperature  changes, 𝛥𝑉  is  measured  at  the  driving  frequency  (first  harmonic)  while  to  measure  Joule 

heating 𝛥𝑉  is measured at twice this frequency (second harmonic).  

 

During scanning thermovoltage (scanning thermal gate microscopy, SThGM) experiments, the same setup is 

used but the AC bias on the Wheatstone bridge has a  lower frequency (57 Hz). Since our probe has a time 

constant  below  1ms,  one  order  of  magnitude  faster  than  the  AC  frequency,  it  is  ensured  that  we  reach 

thermal equilibrium throughout each AC cycle. The temperature generated in the probe heater varies from 0 

to  40  K.  Tovee et  al.  have  shown  that  low  thermal  conductivity materials  (<  2‐5 W (m k)‐1)  can  get  locally 

heated by a hot tip.7 To quantify this heating and obtain a local Seebeck coefficient, we need to estimate the 

9

temperature  at  the  tip‐sample  contact.  As we  know  the  cantilever  thermal  resistance we  can  deduce  the 

measured thermal resistance between the heater and the sample 𝑅 . The latter is the sum of 3 components: 

the thermal resistance of the silicon conical tip 𝑅 , the interface resistance between the tip and the sample 

𝑅  and  the  thermal  spreading  resistance  within  the  sample 𝑅 .  Estimate  of 𝑅  are  obtained  from 

analytical  formula of a circular heat  source on a  layer on substrate.8,9 𝑅  and 𝑅  are  relatively unknown 

even if models exist.10 However, as we measure the total resistance 𝑅  between the heater and the sample, 

we can obtain the sum 𝑅 𝑅 𝑅 𝑅  and deduce the temperature at the sample surface from  

𝛥𝑇 𝛥𝑇 .                       (S6) 

With our measurements, we obtained 𝛥𝑇 18 2 K . 

It is worth noting here, that the Peltier heating/cooling source is distributed. Since the Seebeck coefficient is 

spatially dependent on the mean free path (see section 11), a signal is expected as soon as the edge disorder 

and  defects  become  appreciable.  However,  while  the  source  is  distributed,  the  weight  of  Peltier 

heating/cooling  is  not  constant  but  rather  increases  with  a  narrower  constriction  in  accordance  with  the 

sharply decreasing thermopower (see Figure S11). Therefore, the majority of heating/cooling is taking place 

in the narrowest part of the constriction (the reason for a node in the middle of the constriction is that the 

Peltier switches from heating to cooling)  and is decaying as the width of the constriction increases. 

A second factor defining the shape of the signal in the measurements is the spreading of heat in our sample. 

Our SThM measurement on device without self‐heating show that heat dissipation from the heated tip in the 

areas with and without graphene differs by  less  than 5%, with  limited broadening of  the graphene/ SiO – 

SiO boundary (see Figure S6). This suggests that it is reasonable to assume that the SiO /Si substrate plays 

the  dominant  role  in  the  heat  spreading.  This  is  similar  to  previous measurements  that  suggest  that  heat 

spreading in single‐layer graphene on SiO is poor, and in our case below the characteristic dimensions of a 

100nm constriction.11 

10

In  addition, we  can  consider  a phonon mean  free path of  around 100 nm.12 While  graphene  is  decoupled 

from the oxide due to the interface resistance it still interacts strongly with the substrate, which is why the 

thermal  conductivity  (and  the mean  free  path)  are  greatly  reduced.13  Therefore,  as  the  Peltier  heat  gets 

carried away in the lattice by phonons, it interacts with the substrate/impurities/edges over a length scale of 

100 nm or smaller. We observe a  temperature distribution decaying up to 1 μm from the constriction as a 

result of this heat spreading, which is within ten times the mean free path.   

 

 

Figure S6: SThM map of the tip-sample thermal resistance (𝑅 of a device without excitation. The contrast between graphene

and SiO is difficult to make out in the image as the heat dissipation does not change much. In addition, the constriction does not

affect the spreading much either as can be seen from the linecut shown in the inset (yellow line), where the spikes and big

changes are the contacts on the left and right. The scale bar is 1 μm.

 

 

11

8. Finite Elements Analysis 

To compare the modelled Seebeck coefficient distribution (see section 11) and the subsequent Peltier effect 

with  experimental  results,  we  turned  to  Finite  Element  Analysis  (FEA)  using  COMSOL  MULTIPHYSICS©. 

Realistic  3D  models  included  heat  transfer  in  solid,  electrical  current  and  multiphysics  effects,  e.g. 

thermoelectric effects. Dimensions of the models were set large enough (10 µm) compared to the device (< 3 

µm) to avoid any dimensional artefact  in  the results. As  the device  is  resting on a 300 nm silicon oxide on 

silicon, with thermal conductivity of silicon much higher than the one for SiO2 we assumed that  the silicon 

was acting as a thermal bath. In addition, we assumed perfect coupling between the graphene and the oxide 

and predominantly diffusive heat transfer. 

The main Equation system solved for this system is: 

𝜌𝐶 𝒖 𝛻𝑇 𝛻 𝑘𝛻𝑇 𝑄𝛻𝑱 𝑄 ; 𝑱 𝜎𝐸 𝑱𝒆 ; 𝑬 𝛻𝑉

𝜌𝐶 𝒖𝛻𝑇 𝛻 𝑘𝛻𝑇 𝑃𝑱 𝑄                       (S7) 

where 𝜌, 𝐶 , 𝑘, 𝜎  are  the  materials  properties  of  density,  heat  capacity  and  thermal  and  electrical 

conductivities, respectively. 𝑇, 𝑄, 𝑱, 𝑬, 𝑉 are the temperature, heat density, current density, electric field and 

potential, respectively. 

 

12

Figure S7: meshed geometry of the bow tie device (left) and modelled temperature field for a positive DC current (right). The

asymmetric Joule heating seen on the right panel is due to the Peltier effect.

The geometry of the device is known from manufacturing and AFM measurements (Figure S7 a). The Seebeck 

coefficient  modelled  in  the  main  text  is  set  is  for  the  graphene  Seebeck  coefficient  and  is  thus  position 

dependent.  As  the  current  was  recorded  during  measurement,  the  only  unknown  parameters  are  the 

electrical  and  thermal  conductivities  of  the  graphene  layer  (we  assume  1.4  Wm‐1K‐1  for  the  thermal 

conductivity of the silicon oxide). These two parameters will be assumed as fitting parameters to match the 

observed Joule heating as we explain here after.  

 

We proceeded as follows. One side of the bow tie is grounded while to the other side we apply a DC current 

and calculate the resulting temperature field produced in the device. Then we change the polarity of the DC 

current.  These  temperature  fields  include  both  Peltier  and  Joule  effects.  To  separate  them,  we  extract 

modelled temperature profiles from both simulations and define the Peltier and Joule temperatures as 

𝑇 𝑥                                            (S8)

𝑇 𝑥                                     (S9) 

 

 Once we  obtained  a  Joule  heating  temperature  profile  as  in  Figure  S7  b,  we  can  compare  it  to  the  data 

received from the SThM measurements. Using the same current in the FEA as in the experiments, we used 

the  smallest  Joule  heating  value 1 𝑉  (17.2 μA)  to  fit  the  modelled  profiles  to  the  measured  ones  by 

changing the electrical and thermal conductivities of the graphene layer. We obtained a good fit for realistic 

parameters of 120 Wm K and 5 ∙ 10 Sm . These fitting parameters are then used for all subsequent 

calculations such as the Joule heating at higher currents and the Peltier simulations. 

Ground

Source

13

9. Current dependency of the island and ribbon geometries  

The  behavior  of  the  bowtie  devices  at  high  currents,  where  a  change  from  a  linear  to  a  cubic  current 

dependence is observed, is also seen in the other geometries studied. Again, the SThM tip is parked to the 

left or right side of a constriction where the signal due to a geometrical change is perceived highest  in the 

map. Then the current through the device is ramped up as the signal strength is recorded. As shown in Figure 

S8, a combination of a linear and cubic dependence can be fitted to both the island and ribbon geometry, the 

crossover  point  observed  at  somewhat  different  current  values within  50%.  This  is  further  evidence  for  a 

current induced origin of the enhancement of the Peltier effect.  

 

14

 

Figure S8: Current dependence measurements of the Peltier signal for (a) the island geometry and (b) the long ribbon.

 

 

15

10. Long ribbon: 

The thermoelectric effects are prevalent in all geometries that were tested here, such as longer ribbons and 

island  structures  and  is  of  comparable  strength  in  all  of  them.  In  both  the  Peltier  and  thermovoltage 

measurements  (Figure  S9  a  and  b),  a  fine  structure  is  showing,  similar  to  previously  observed  effects  in 

photocurrent nanoscopy of graphene that are attributed to charge puddles, defects and inhomogeneities,14 

in addition to the edge disorder prevalent in our samples due to the oxygen plasma fabrication step. 

 

Figure S9: Thermoelectric measurements on the wire geometry. (a) Lateral Friction image of the wire geometry. (b) Peltier map

of the wire geometry. Multiple sign changes in the middle of the ribbon are visible, in addition to a defect caused signal on the

right. (c) Thermovoltage map of the wire geometry. Again, a strong signal is visible on the right-hand side of the device due to a

defect and a fine structure is visible that is shown again in a close-up.

The scale bars are 500 nm in (a) and (c) in the main measurement, 250 nm in (b) and 100 nm in the close up in (c).

16

In  all measurements,  there  is  a  stark  effect  at  the boundary between bulk  graphene and  the  sub 100 nm 

constriction, reinforcing the notion that the constriction itself is responsible for the effect. In order to show 

the high resolution and the fine structure  in our measurements, a  long graphene ribbon  is  investigated.  In 

addition to a defect on the side that is causing a big signal, multiple sign changes and fine structures can be 

observed.   

11. Spatially dependent Seebeck coefficient  

We will start by giving a definition of electrical current in the linear response regime under an electric field or thermal 

bias. For electrical current density 𝒋, the linear response in terms of the electric field 𝑬 and the temperature gradient Δ𝑇 

is given by 

𝒋 𝜎𝑬 𝜎𝑆Δ𝑇 .                                    (S10) 

where 𝜎 is the electrical conductivity, and 𝑆 is the Seebeck coefficient describing the thermopower. We consider current 

along the 𝑥 direction, 𝑬 𝑑𝑉/𝑑𝑥, and: 

𝑗 𝜎 𝜎𝑆  .                                   (S11) 

In the following, we explain how to calculate the spatial dependence of the Seebeck coefficient for a bowtie device by 

first  considering  its  resistance. We begin with  the diffusive  conductance 𝐺 𝑘 𝑙  of  a  rectangular  graphene 

device of length 𝐿 in the longitudinal 𝑥 direction and width 𝑊 in the transverse 𝑦 direction with mean free path 𝑙. In a 

bowtie geometry  (shown in Figure S9),  15–17  the width of the device  in the 𝑦 direction varies  from a maximum 𝑌

2𝑡 𝑊 to a minimum𝑊.  

17

 

Figure S10. Schematic of the bowtie junction of length 𝐿, maximum width 𝑌, minimum width 𝑊 and constriction width 

y(x).  

To determine the spatially‐dependent resistance 𝑅 𝑥  and spatially‐dependent Seebeck coefficient 𝑆 𝑥), we consider a 

thin strip of infinitesimal length 𝑑𝑥 located at length 𝑥 from the centre of the constriction (where 𝑥 0) and of width 

𝑦 𝑥 : on the right hand side, 0 𝑥 𝐿/2 and 𝑊 𝑦 𝑥 𝑌. 

We take into account two spatially‐dependent effects: (i) a geometrical effect whereby the width change as a function of 

𝑥 is taken into account by replacing 𝑊 in the diffusive conductance; (ii) a spatially dependent mean free path 𝑙 𝑥  to 

model scattering at edges which is relevant near the constriction. 

In  particular, we  consider  an  infinitesimal  strip  of  length 𝑑𝑥 located  at  position 𝑥 on  the  right  hand  side 𝑥 0,  as 

depicted in Figure S10. The corresponding width 𝑦 𝑥  at that point is 

𝑦 𝑥 4𝑡𝑥/𝐿 𝑊; 0 𝑥 𝐿/2; 𝑊 𝑦 𝑥 𝑌.                          (S12) 

 

This can then be used to write the conductance as  

18

𝐺 𝑥 𝑘 𝑙 𝑥  ,                                   (S13) 

where 𝑙 𝑥  is a position dependent mean free path. We model it generically as 

𝑙 𝑥 𝑙 1 𝑐                                   (S14) 

where 𝑙  is the bulk mean free path in our graphene devices and 𝑛 and 𝑐  are real numbers.  

Thus, the contribution of the graphene strip to the resistance is  

𝑅 𝑥 .                                   (S15) 

To determine the total resistance, we sum resistances in series by integrating with respect to 𝑥. Note that the EMFP is 

only explicitly  affected by  scattering  in  the  transverse direction, hence  the dependence on  the width y(x)  in  Eq.(1). 

However, for the bow tie, y(x) is itself a function of x by geometry and, as we sum resistances in series in our derivation, 

we  change  variables  from  y(x)  to  x  in  order  to  integrate with  respect  to  the  variable  x.  The  left  hand  side  of  the 

constriction gives the same contribution as the right, giving a factor of two, and the total resistance is  

𝑅 1 𝑐   

𝑑𝑦 𝑐

ln   

ln                                 (S16) 

where,  in the  last  line, 𝑌 ≫ 𝑊 was used. The first term is similar to the diffusive  (Drude) resistance 𝐺  introduced 

earlier for a rectangular system and, for 𝑛 1 and 𝑐 1, the second term, due to ‘boundary’ scattering, is  

𝑅                                     (S17) 

19

In other words, the total resistance is approximately equal to a diffusive and a boundary scattering contribution.18 This 

enables us to calculate the Seebeck coefficient by writing the Mott formula (𝑆

| ) as  

  𝑆

| ,                                 (S18) 

with  

𝑅 𝜖 ℏ

ℓln , where we used 𝜖 ℏ𝑣𝑘. 

We then find that  

𝑆 .                                              (S19) 

Going back to the resistance of a strip in Equation S15,  

𝑅 𝑥ℏ

ℓ𝑐 𝑑𝑥,                                                 

(S20) 

we can determine local thermopower via Equation S18: 

𝑆 𝑥 1 𝑛 𝑈 𝑛 1 𝑈 ,                               (S21) 

where 𝑈 | . This means that our Seebeck coefficient effectively changes due to the changing width in our 

constriction (see Figure S11), explaining the signal that is observed.  

20

 

FigureS11. Calculated EMFP and Seebeck coefficient S for the bowtie constriction due to increased edge scattering for the fitting

parameters obtained from Figure 2, giving a bulk mean free path of l 226 nm and S 118 μVK .

Using this model with 𝑛 2.6 and 𝑐 0.56 and a bulk mean free path 𝑙 226 nm the EMFP is reduced 

substantially within our constriction region to about 40 𝑛𝑚 as shown below in Figure S10. This in turn reduces the 

Seebeck coefficient from  118 𝜇𝑉𝐾  to  0.35 𝜇𝑉𝐾  in our model, a change over 3 orders of magnitude. This is 

due to Equation (S21) involving a difference of terms; rather than a seemingly linear dependence on the mean free path, 

a small change in the EMFP can then result in a vastly differing Seebeck coefficient. It is also worth noting here, that for 

an almost ballistic sample where 𝑛 1 and 𝑙 ≫ Δ𝑦 the dependence of S on the width is just a linear one: 𝑆 1

but our fitting shows that we are in a diffusive/ballistic crossover regime instead. The bulk mean free path is 

extracted from gate dependent measurements of long ribbons as discussed earlier in section 3 and the fitting 

parameters 𝑛 and 𝑐 are obtained from a thermovoltage fit as explained in Figure 2. 

 

21

12. Gate dependency 

Through  back  gating  of  our  devices,  different  carrier  concentrations  can  be  induced,  shifting  the  doping 

from 𝑝  to𝑝.  

 

Figure S12. Gate dependent thermovoltage measurements. Global thermovoltage recorded as the hot tip is scanned over a bow

tie device at two different back gate voltages 𝑉 30𝑉 and 𝑉 30𝑉.

The scale bar is 500 nm.

As  can  be  seen  in  Figure  S12,  gating  the  sample  does  change  the  strength  and  distribution  outside  the 

constriction,  however  the  general  signal with  a positive  thermovoltage on  the  left  side,  a negative on  the 

right and a node in the constriction stays the same. The change in strength and shape can be attributed to a 

change in the bulk Seebeck coefficient due to the varying carrier densities.19 

 

 

13. Movement of heat by high charge carrier drift velocities  

 

A possible mechanism of nonlinearity and cause of the asymmetry observed in the heat generation originates 

in the high drift velocity causing charge carriers to carry heat in a non‐symmetric fashion.  

For a drift velocity 𝑣  much smaller than the Fermi velocity 𝑣  i.e. 𝑣 ≪ 𝑣 , energetic hot charge carriers are 

dispersing from the middle of the constriction in an approximately radial pattern. 

22

Each of the electrons carries a thermal energy of kBT and travels the mean‐free‐path (MFP) distance 𝑙 𝜏𝑣  

on average (where 𝜏 is the relaxation time) before depositing its thermal energy within the graphene lattice, 

in a circle around the highest temperature spot (Figure S13 a and b).  

 

 

 

Figure S13. Illustration of the heat blowing effect due to the high drift velocity in a nanoconstriction. (a) schematic of Joule

heating at low currents in a bow-tie device. (b) Schematic of Joule heating at high currents in a bow-tie device, showing the effect

the shifting of the heat has. (c) Close up on the heat spot due to Joule heating at low currents in a bow-tie device. (d) Close up on

the heat spot due to Joule heating at high currents in a bow-tie device, showing the effect the shifting of the heat shifting with the

current direction. 

 

J

MFP MFP

MFP = vF

(VF -

VdVd

b d

a c

(VF+Vd)

W

23

In our experiments the current through the device I was up to 85 𝜇𝐴 resulting in a drift velocity of 𝑣

0.499 ⋅ 10 ms , which  is  on  the  same  order  as  the  Fermi  velocity  of  10 ms .  As  the  drift 

velocity  increases  and  becomes  non‐negligible  compared  to  the  Fermi  velocity,  the  area  where  the  hot 

electrons are depositing heat is shifted in the direction of the drift velocity by the distance 𝑣 𝜏 (Figure S13 c 

and d).  

 

This heat produces a temperature difference along the direction of the current flow a distance 2𝑣 𝜏 apartIt 

should  be  noted  that  if  the  channel  length  l  is much  longer  than 2𝑣 𝜏 only  the  entry  and  the  exit  to  the 

channel  will  have  this  response  linked  to  the  drift  velocity.  This  was  confirmed  in  our  measurements  of 

graphene  islands  and  a  long  ribbon,  suggesting  that  this  current  drifting  effect  on  Joule  heating  is  most 

pronounced for hot spots in small constrictions. 

This makes it clear, that we do not expect a linear dependence of the Peltier heating/cooling on the current 

in  our  experiment  as  one might  conclude  from 𝑄 Π𝐼.  Due  to  the  strong  Joule  heating  induced  by  the 

current, the measured signal seems to change regimes as the current is increased.  

Furthermore, we tested other higher order terms in addition to a cubic dependence. As can be seen in Figure 

S14  4th  and  5th  order  terms overestimate  the  values  at  lower  bias while  a  quadratic  term underestimates 

them. However, while the cubic fit is the best fit it should be noted that the other fits lie mostly within the 

24

error  bars  and  cannot  be  ruled  out  categorically. 

 

Figure S14. Different fitting parameters for the Peltier signal current dependency. (a) Linear and 5th order current dependency fit

to the Peltier heating. The orange line is linear with respect to the current and the blue line is proportional to the current to the

power of 5. (b) Linear and Cubic (grey), 5th order (red), quadratic (red dashed) or 4th order (red dotted) dependency of the Peltier

signal on current compared to the experimental data (green dots). 

 

25

14. Island geometry measurements 

To  test  the  hypothesis  that  a  geometrical  effect  causing  a  reduction  in  the  EMFP  is  responsible  for  the 

observed  signal  and  to  rule  out  the  transport  of  hot/cold  carriers  from  the  Au/graphene  junction  at  the 

electrodes,  we  investigated  a  “multi‐island”  geometry  (see  Figure  S15).  Here,  wide  areas  alternate  with 

constrictions, thus giving us a better control over the change from bulk to edge‐scattering dominated regions 

and low to high current density. 

Figure S15. (a) Lateral Friction image of the island geometry. A crack in the graphene at the left side creates an additional

constriction in this device. (b) Thermovoltage map of the island geometry. A pronounced signal including a sign change in the

middle of each intact ribbon is visible and a fine structure in the signal can be seen that is investigated further (see SI). (c) Peltier

map of the island geometry. A signal is visible within each ribbon, confirming, that the effect measured is caused by a

constriction-induced change in then Seebeck coefficient. The asymmetric signal that deviates slightly from the thermovoltage

signal is discussed in the main text. The scale bar is 500 nm in all figures.

 

26

As  can  be  seen  in  Figure  S15,  the  island  geometry  shows  heating/cooling  pattern  and  positive/negative 

thermovoltage  at  the  constriction  to  bulk  interfaces  in  accordance with  the  observed  spatially  dependent 

Seebeck coefficient due to  increased edge scattering.  In addition,  the  signal within  the  ribbons of  reduced 

length  shows  small  variations,  which  demonstrates  the  high  susceptibility  of  the  Seebeck  coefficient  to 

defects, charge puddles and small changes in width that were seen previously in photocurrent nanoscopy.14 

This sensitivity to small variations of the Seebeck coefficient inside a narrow graphene channel was further 

studied in long ribbons (see Figure S9), highlighting the resolution of our technique, which is able to resolve 

sub 20 nm features in the thermovoltage measurements. 

 

15. Sample preparation 

In order  to ensure  the high quality and cleanliness of  the  samples,  they were vacuum annealed at 300  ◦C 

prior to measurement to remove PMMA and hydrocarbon contaminants.20 Finally, the AFM tip was heated 

and quickly scanned over the sample, to completely remove remaining polymer residues naturally present on 

electron‐beam fabricated samples prior to our measurements. The samples were at this point already loaded 

in  the measurement  SThM  vacuum  chamber.  All  devices  are  inherently  p‐doped with  a  charge  neutrality 

point  at  40‐60V  (see  Figure  S2),  due  to  oxygen  rich  open‐shell/dangling  bond  defects  and  hydrocarbon 

surface contamination as observed previously.21,22 

16. Peltier signal at the contacts 

In addition to the Peltier signal within the constriction, a weaker signal can be observed from the contacts that lies just above the noise level. As can be seen in Figure S16 where the Peltier signal of device 2 is shown with a saturated color scale, there are stripes or heating on the left side and stripes of cooling on the right side at the position of the contacts in addition to a slightly warmer and colder area in front of the contacts. The reason for this signal is from band bending due to doping of graphene via the metal contacts as observed previously.23,24 Importantly, the signal from the contacts is substantially smaller than the one observed in the constriction. 

27

 

Figure S16. Saturated colorscale Peltier temperature map of device 2. While there is a rather high noise level, Peltier heating/cooling at the contacts can be observed in addition to the signal in the constriction.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

28

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